FACTORIZATION IN MONOIDS AND RINGS SALVATORE TRINGALI Abstract. Let H× be the group of units of a multiplicatively written monoid H. We say H is acyclic if xyz 6= y for all x,y,z ∈ H with x∈ / H× or z∈ / H×; unit-cancellative if yx 6= x 6= xy for all x,y ∈ H with y∈ / H×; f.g.u. if there is a finite set A ⊆ H such that every non-unit of H is a finite product of elements of the form uav with u, v ∈ H× and a ∈ A; l.f.g.u. if, for each x ∈ H, the smallest divisor-closed submonoid of H containing x is f.g.u; and atomic if every non-unit can be written as a finite product of atoms, where an atom is a non-unit that does not factor into a product of two non-units. We generalize to l.f.g.u. or acyclic l.f.g.u. monoids a few results so far only known for unit-cancellative l.f.g.u. commutative monoids (cancellative monoids are unit-cancellative, and a commutative monoid is unit-cancellative if and only if it is acyclic). In particular, we prove the following: • If H is an atomic l.f.g.u. monoid, then every non-unit has only finitely many factorizations (into atoms) that are “minimal” and “pairwise non-equivalent” (with respect to some naturally defined relations on the free monoid over the “alphabet” of atoms). • If H is an acyclic l.f.g.u. monoid, then it is atomic; and moreover, each element has only finitely many “pairwise non-equivalent” factorizations if we additionally assume H to be commutative. 1. Introduction By the fundamental theorem of arithmetic, every positive integer other than 1 can be expressed as a non-empty product of primes in an essentially unique way. Factorization theory is, on the whole, the study of various phenomena related to the possibility or impossibility of extending such a decomposition to arbitrary rings and monoids (see § 2 for basic terminology). Over the years, the field has branched out into many subfields, ranging from commutative and non- commutative algebra to semigroup theory, from additive combinatorics to the abstract theory of zeta functions. Detailed information about problems, methods, results, and trends can be found in the con- ference proceedings [2, 14, 12, 13], in the surveys [9, 8, 23, 27], and in the monographs [33, 24, 21]. So far, research in the area has been centered on rings and monoids — e.g., monoids of modules, Mori domains, Krull monoids, rings of integer-valued polynomials, monoids of ideals, orders in central simple algebras, monoids of matrices, and numerical monoids —, where the structures in play are commutative arXiv:2005.01681v1 [math.RA] 4 May 2020 or cancellative. It is only recently [19, 4] that some key aspects of factorization theory have been system- atically extended to possibly non-commutative and non-cancellative monoids (and thence to rings that need not be domains), so as to widen the spectrum of potential applications and foster interaction with other fields. In the present paper, we further contribute to this line of research. In particular, we say that a monoid H is atomic if every non-unit element of H is a product of atoms (Definition 2.2); l.f.g.u. if, for every x ∈ H, the smallest divisor-closed submonoid of H containing x is, 2010 Mathematics Subject Classification. Primary 20M10, 20M13. Secondary 13A05, 16U30, 20M14. Key words and phrases. Acyclic monoids; atomic monoids; BF-monoids; FF-monoids; fundamental theorem of arith- metic; minimal factorizations; primes; unique and non-unique factorization. 2 Salvatore Tringali up to units, finitely generated (Definition 3.1); and acyclic if certain “cyclic relations” are forbidden in H (Definition 4.1). Among others, we will prove the following results, so far only known for commutative and “nearly cancellative” monoids (and rings): • An atomic l.f.g.u. monoid is FmF (Corollary 3.5), that is, every non-unit has only finitely many factorizations into atoms that are “minimal” and “pairwise non-equivalent” with respect to some naturally defined relations on the free monoid over the “alphabet” of atoms. • An acyclic l.f.g.u. monoid is atomic (Corollary 4.12), and is actually FF (meaning that each ele- ment has only finitely many “pairwise non-equivalent” factorizations into atoms) if we additionally assume that the monoid is commutative (Corollary 4.13). In addition, we will establish a characterization of “unique factorization monoids” (Theorem 5.9) in terms of a special type of atoms we call powerful (Definition 5.5), whose relation with primes (Definition 5.1) and the fundamental theorem of arithmetic is clarified by Propositions 5.3 and 5.8 and Example 5.10. The paper closes with some ideas for further research (§ 6) and includes many stubs that shall help, we hope, to shed light on some delicate points (e.g., Examples 4.14, 4.15, 5.4, and 5.7). 2. Preliminaries. In this section, we establish some notations and terminology used all through the paper and prepare the ground for the study of l.f.g.u. and acyclic monoids in §§ 3 and 4. Further terminology and notations, if not explained when first introduced, are standard or should be clear from context. 2.1. Generalities. We use N for the non-negative integers, Z for the integers, Q for the rationals, and R for the reals. For all a,b ∈ R, we let Ja,bK := {x ∈ Z : a ≤ x ≤ b} be the discrete interval between a and b. Unless noted otherwise, we reserve the letters m and n (with or without subscripts) for positive integers, and the letters i, j, k, and ℓ for non-negative integers. Given a set X and an integer k ≥ 0, we write X×k for the Cartesian product of k copies of X; and if R is a binary relation on X and x is an element of X, we let JxKR := {y ∈ X : xRy}⊆ X, where xRy is a shorthand notation for (x, y) ∈ R. A (partial) preorder on X is a binary relation R on X such that x R x for all x ∈ X (i.e., R is reflexive), and xRz whenever xRy and yRz (i.e., R is transitive). An order on X is a preorder R on X such that, if xRy and y R x, then x = y; and an equivalence (relation) on X is a preorder R on X such that xRy if and only if y R x. If R is a preorder (resp., an order) on X, then we call the pair (X, R) a preset (resp., a poset). We will commonly denote a preorder by the symbol (with or without subscripts or superscripts), and reserve the symbol ≤ (and its “dual” ≥) for the usual order on the reals and its subsets. Accordingly, we shall write x ≺ y if x y and y 6 x; notice that “x ≺ y” is a stronger condition than “x y and x 6= y”, and the two conditions are equivalent when is an order. If (X, ) is a preset and Y is a subset of X, we let a -minimal element of Y be an element y¯ ∈ Y with the property that there exists no element y ∈ Y such that y ≺ y¯. 2.2. Monoids. Throughout, monoids will be usually written multiplicatively and, unless a statement to the contrary is made, need not have any special property (e.g., commutativity). × Let H be a monoid with identity 1H . We denote by H and A (H), resp., the set of units and the set of atoms of H, where u ∈ H is a unit if uv = vu =1H for some provably unique v ∈ H, called the inverse On the Arithmetic of Monoids and Rings 3 of u (in H) and denoted by u−1; and a ∈ H is an atom if a∈ / H× and a 6= xy for all x, y ∈ H r H×. It is easily seen that H× is a subgroup of H, hence referred to as the group of units of H. A (monoid) congruence on H is an equivalence relation R on H such that, if xRu and y R v, then xy R uv. Note that, if R is a congruence on H, we will usually write x ≡ y mod R in place of xRy. × We say H is reduced if H = {1H }; cancellative if xz = yz or zx = zy, for some x,y,z ∈ H, implies x = y; and unit-cancellative if xy 6= x 6= yx for all x, y ∈ H with y∈ / H×. Unit-cancellative monoids have been the subject of several recent papers in factorization theory, both in the commutative [18, 25] and in the non-commutative setting [26, 19, 4]; and among others, it is obvious that a cancellative monoid is unit-cancellative (see also Remark 4.2). n + Given x ∈ H, we call ordH (x) := |{x : n ∈ N }| ∈ N ∪ {∞} the order of x (relative to H); and we 2 say that x is an element of finite order (in H) if ordH (x) < ∞, an idempotent if x = x, and a non-trivial 2 idempotent if x = x 6=1H . For all X1,...,Xn ⊆ H, we denote by X1 ··· Xn := {x1 ··· xn : x1 ∈ X1,...,xn ∈ Xn}⊆ H the setwise product of X1,...,Xn (relative to H); and by abuse of notation we replace Xi with xi on the left of the symbol “:=” in the above definition if Xi = {xi} for some i ∈ J1,nK and there is no risk of confusion.